Lift factors for the Nielsen root theory on n-valued maps - Brown, Robert F.; Gonçalves, Daciberg Lima (Topological Methods in Nonlinear Analysis, 2023)
A root of an n-valued map φ X → D_n(Y) at a Y is a point x X such that a φ(x). We lift the map φ to a split n-valued map of finite covering spaces and its single-valued factors are defined to be the lift factors of φ. We describe the relationship between the root classes at a of the lift factors and those of φ. We define the Reidemeister root number (φ) in terms of the Reidemeister root numbers of the lift factors. We prove that the Reidemeister root number is a homotopy invariant upper bound for the Nielsen root number NR(φ), the number of essential root classes, and we characterize essentiality by means of an equivalence relation called the Φ-relation. A theorem of Brooks states that a single-valued map to a closed connected manifold is root-uniform, that is, its root classes are either all essential or all inessential. It follows that if Y is a closed connected manifold, then the lift factors are root-uniform and we relate this property to the root-uniformity of φ. If X and Y are closed connected oriented manifolds of the same dimension then, by means of the lift factors, we define an integer-valued index of a root class of φ that is invariant under Φ-relation and this implies that if its index is non-zero, then the root class is essential.